Abstract

We derive some results for a new class of analytic functions defined by using Salagean operator. We give some properties
of functions in this class and obtain numerous sharp results including
for example, coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, integral means
inequalities, and partial sums of functions belonging to this class. Finally, we give an application involving certain fractional calculus operators that are also considered.

1. Introduction

Let denote the class of functions of the form
that are analytic and univalent in the open unit disc .

Definition 1 (subordination principle). For two functions and , analytic in , we say that the function is subordinate to in and write , if there exists a Schwarz function , which (by definition) is analytic in with and , such that . Indeed it is known that
Furthermore, if the function is univalent in , then we have the following equivalence [2, page 4]:

Definition 2 (see [3]). Let denote the subclass of consisting of functions of the form (1) and satisfy the following subordination:
Specializing the parameters , and , we obtain the following subclasses studied by various authors:(i)
(see Eker and Owa [4]);(ii)
(see Shams et al. [5, 6]);(iii)
(see Janowski [7] and Padmanabhan and Ganesan [8]).

Also we note that

Let denote the subclass of functions of of the form

Further, we define the class by

For suitable choices of the parameters , and , we can get various known or new subclasses of . For example, we have the following:(i) (see Rosy and Murugusundaramoorthy [9] and Aouf [10]);(ii) and (see Bharati et al. [11]);(iii) and (see Silverman [12]).

2. Coefficient Estimates

Unless otherwise mentioned, we assume in the reminder of this paper that and .

Now, we will need the following lemma which gives a sufficient condition for functions belonging to the class .

In Theorem 4, it is shown that the condition (12) is also necessary for functions of the form (10) to be in the class .

Theorem 4. Let . Then if and only if

Proof. In view of Lemma 3, we only need to prove the only if part of Theorem 4. Since , for functions , we can write
then
Since , then we obtain
Now choosing to be real and letting , we obtain
Or, equivalently
This completes the proof of Theorem 4.

Remark 5. (i) The result obtained by Theorem 4 corrects the result obtained by Li and Tang [3, Theorem 1].(ii) Putting and in Theorem 4, we correct the result obtained by Eker and Owa [4, Theorem 2.1].(iii) Putting , and in Theorem 4, we obtain the result obtained by Rosy and Murugusudaramoorthy [9, Theorem 2].

Corollary 6. Let the function be defined by (10) and let it be in the class . Then
The result is sharp for the function

3. Distortion Theorems

Theorem 7. Let the function defined by (10) be in the class . Then
The result is sharp.

Proof. In view of Theorem 4, since
is an increasing function of , we have
that is
Thus we have
Similarly, we get
Finally the result is sharp for the function
at and . This completes the proof of Theorem 7.

Theorem 8. Let the function defined by (10) be in the class . Then
The result is sharp.

Proof. Similarly is an increasing function of , in view of Theorem 4, we have
that is
Thus we have
Similarly
Finally, we can see that the assertions of Theorem 8 are sharp for the function defined by (27). This completes the proof of Theorem 8.

4. Radii of Starlikeness, Convexity, and Close-to-Convexity

In this section radii of close-to-convexity, starlikeness, and convexity for functions belonging to the class are obtained.

Theorem 9. Let the function defined by (10) be in the class ; then(i) is starlike of order in , where
(ii) is convex of order in , where
(iii)is close-to-convex of order in , where
Each of these results is sharp for the function given by (20).

Proof. It is sufficient to show that
where is given by (33). Indeed we find from (10) that
Thus we have
if and only if
But, by Theorem 4, (39) will be true if
that is, if
Or
This completes the proof of (33).

5. Extreme Points

Theorem 10. Let
Then if and only if it can be expressed in the following form:
where

Proof. Suppose that
Then, from Theorem 4, we have
Thus, in view of Theorem 4, we find that .Conversely, let us suppose that , then, since
Set
Thus clearly, we have
This completes the proof of Theorem 10.

6. Integral Means Inequalities

Lemma 12. If the functions and are analytic in with
then for and ,
We now make use of Lemma 12 to prove Theorem 13.

Theorem 13. Suppose that ,, and is defined by
Then for , we have

Proof. For , (55) is equivalent to prove that
By applying Littlewood’s subordination lemma (Lemma 12), it would suffice to show that
By setting
and using (13), we obtain
This completes the proof of Theorem 13.

7. Partial Sums

In this section partial sums of functions in the class are obtained, also we will obtain sharp lower bounds for the ratios of real part of to .

Theorem 14. Define the partial sums and by
Let the function be given by (1) and let it satisfy the condition (12) and
where, for convenience,
Then

Proof. For the coefficients given by (64) it is not difficult to verify that
Therefore we have
By setting
and applying (68), we find that
Now
if
From the condition (12), it is sufficient to show that
which is equivalent to
which readily yields the assertion (65) of Theorem 14. In order to see that
gives sharp result, we observe that for that as . Similarly, if we take
and making use of (68), we can deduce that
which leads us immediately to the assertion (66) of Theorem 14.The bound in (66) is sharp for each with the extremal function given by (75). Then the proof of Theorem 14 is completed.

8. Distortion Theorems Involving Fractional Calculus

In this section, we will prove several distortion theorems for functions belonging to the class . Each of these theorems would involve certain operators of fractional calculus (i.e., fractional integrals and fractional derivatives), which are defined as follows (see, for details, [15–18]). For our present investigation, we recall the following definitions.

Definition 15. The fractional integral of order is defined, for a function , by
where the function is analytic in a simply connected domain of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .

Definition 16. The fractional derivative of order is defined, for a function , by
where the function is constrained, and the multiplicity of is removed as in Definition 15.

Definition 17. Under the hypotheses of Definition 16, the fractional derivative of order is defined, for a function , by
Using Definitions 15, 16, and 17, we obtain
in terms of Gamma functions.

Theorem 18. Let the function defined by (10) be in the class . Then
The results are sharp.

Proof. Let
where
Since is a decreasing function of , we can write
Furthermore, in view of Theorem 4, we have
Then
Therefore, by using (85) and (87), we can see that
and similarly
which prove Theorem 18.Finally, the equalities are attained for the function defined by
or, equivalently, by given by (27).Then the results are sharp, and the proof of Theorem 18 is completed.

Corollary 19. Under the hypothesis of Theorem 20, is included in a disk with its center at the origin and radius given by

Theorem 20. Let the function defined by (10) be in the class . Then
Each of these results is sharp.

Proof. Let
where
Since is a decreasing function of , we can write
Furthermore, in view of Theorem 4, we have
Then
Therefore, by using (95) and (97), we can see that
and similarly
which together prove the two assertions of Theorem 20.Finally, the equalities are attained for the function defined by
or, equivalently, by given by (27).Then the result is sharp, and the proof of Theorem 20 is completed.

Corollary 21. Under the hypothesis of Theorem 20, is included in a disk with its center at the origin and radius given by

Acknowledgment

The authors thank the referee for his valuable suggestions which led to improvement of this study.